Question

A very nice (an interesting) formula for the entropy of a system is S=−k∑=1p(ni) lnp(ni), where p(n) is the probability distribution. a. Show this equation holds for a binary system (N=2) using the formula for this multiplicity: Ω =N!/N1!N2! (hint: use Stirling’s approximation).b.Now use this equation and the equation for the probability distribution for a canonical ensemble to show that the entropy can be related to the partition function by S=k∂/∂T[TlnZ].

Answer 1

We know that ...............................(1)

We know that, , putting the value of Ni in the equation (1)

Taking log on both sides

using Stirling's formula

Now multiplying with k and dividing by N to get the entropy per particle.

Hence it proved this interesting relation of entropy and probability.

Now the partition function Z is defined as

And Helmholtz free energy is defined as

Taking the partial derivative of this equation with respect to T, we get

Hence it proof of the relation between the entropy and partition function.

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